Question i. Sketch the curve ii. The region enclosed by the curve, the x-axis and the y-axis is rotated through 360◦ about the x-axis. Find the volume obtained, giving your answer in terms of . Solution i. Standard form of quadratic equation is; The graph of quadratic equation is a parabola. If (‘a’ is […]
Question i. Express in the form and hence state the coordinates of the minimum point, A, on the curve . The line intersects the curve at points P and Q. It is given that the coordinates of P are (3,7). ii. Find the coordinates of Q. iii. Find the equation of the line joining Q to the mid-point of […]
Question In the diagram, OAB is an isosceles triangle with and angle radians. Arc PST has centre O and radius , and the line ASB is a tangent to the arc PST at S. i. Find the total area of the shaded regions in terms of and . ii. In the case where and , find the total perimeter […]
Question A curve is such that and the point lies on the curve. i. Find the equation of the curve. ii. Find the set of values of x for which the gradient of the curve is less than . Solution i. We can find equation of the curve from its derivative through integration; For the given case; Therefore; Rule […]
Question The variables x, y and s can take only positive values and are such that and i. Show that . ii. Find the stationary value of and determine its nature. Solution i. We are given that; We are also given that; We can find out y from this equation; Substituting this value of y in; ii. […]
Question i. Show that the equation can be written in the form ii. Hence solve the equation for . Solution i. We have the equation; We have the relation , therefore, Multiplying entire equation with We have the trigonometric identity; We can rewrite it as; Therefore; ii. To solve the equation , […]
Question The diagram shows a prism ABCDPQRS with a horizontal square base APSD with sides of length 6 cm. The cross-section ABCD is a trapezium and is such that the vertical edges AB and DC are of lengths 5 cm and 2 cm respectively. Unit vectors , and are parallel to AD, AP and AB respectively. i. Express each of […]
Question The volume of a spherical balloon is increasing at a constant rate of 50 cm3 per second. Find the rate of increase of the radius when the radius is 10 cm. [Volume of a sphere ] Solution i. We are given that; We are required to find rate of change of radius; Therefore; For rate of change of […]
Question Functions and are defined for by i. Find and simplify expressions for and . ii. Hence find the value of a for which . iii. Find the value of for which . iv. Find and simplify an expression for . The function is defined by , v. Find an expression for . Solution i. We have functions; We can write these […]
Question Find the coefficient of in the expansion of . Solution Expression for the Binomial expansion of is: In the given case: Hence; The coefficient of is .
Question A television quiz show takes place every day. On day 1 the prize money is $1000. If this is not won the prize money is increased for day 2. The prize money is increased in a similar way every day until it is won. The television company considered the following two different models for increasing the prize money. […]